On stability, convergence and accuracy of bES-FEM and bFS-FEM for nearly incompressible elasticity

TL;DR

This paper develops a rigorous theoretical framework demonstrating that improved edge-based and face-based smoothed finite element methods (bESFEM and bFS-FEM) are stable, convergent, and accurate for nearly-incompressible elasticity problems, avoiding locking and oscillations.

Contribution

It introduces enriched polynomial spaces with bubble functions for these methods, ensuring stability and convergence for nearly-incompressible elasticity.

Findings

Methods satisfy uniform inf-sup condition

Numerical examples confirm high accuracy and stability

Meshes can be generated automatically

Abstract

We present in this paper a rigorous theoretical framework to show stability, convergence and accuracy of improved edge-based and face-based smoothed finite element methods (bESFEM and bFS-FEM) for nearly-incompressible elasticity problems. The crucial idea is that the space of piecewise linear polynomials used for the displacements is enriched with bubble functions on each element, while the pressure is a piecewise constant function. The meshes of triangular or tetrahedral elements required by these methods can be generated automatically. The enrichment induces a softening in the bilinear form allowing the weakened weak (W2)procedure to produce a high-quality solution, free from locking and that does not oscillate. We prove theoretically that both methods confirm the uniform inf-sup and convergence conditions. Four numerical examples are given to validate the reliability of the bES-FEM and bFS-FEM.